CIs and p-values
Much of the inference part of conventional intro stats is about computing confidence intervals and p-values in a variety of settings:
- difference of means
- difference of proportions
- slope of regression line
Formulas from Triola
No statistics were harmed in the filming of this documentary
In the following, spread is measured using the length of the 85% summary interval.
You can still do all this by using the standard deviation instead.
Star notation
The usual notation for statistical significance is something like p < 0.04.
In regression reports, statistical significance is appreviated with stars:
- ★ p < .05
- ★★ p < .01
- ★★★ p < .001
We want at the same time to be able
- To satisfy the traditionalists.
- To put “significance” on a scale that doesn’t lead to familiar but fallacious probability interpretations, e.g. “The probability that the Null is true is 0.01.”
- To make it clear that 0.05 is a weak standard for significance.
- To remind people that things other than “statistical significance” are important.
My proposal: Amazon-like ratings
- No stars: anecdotal, no sampling plan

- One star: sampling plan and p < 0.05

- Two stars: : one star + p < 0.01

- Three stars: two stars + p < 0.001

- Four stars: three stars and covariates considered

- Five stars: four stars and effect size reaches a magnitude of practical importance.

The procedure
- Plot out data, draw in model.
- From data and model, find R and sample size n. The picture will tell you R.
- Measure effect size, e.g. difference in means or slope of regression line. Call it \(\Delta\).
- Calculate \(F = n-2 \frac{R^2}{1-R^2}\).
- 95% confidence interval on \(\Delta\) is \(\Delta \pm \sqrt{4/F}\).
- Old-timers call it “significance”, but let’s call it … ?
- \(F = 4\) – one star (corresponds to about 0.05)
- \(F = 7\) – two stars (corresponds to about 0.01)
- \(F = 12\) – three stars (corresponds to about 0.001)
The F table
Maybe blue, red, white, yellow

Calculating F
## Warning: Removed 8 rows containing missing values (geom_path).

## Scale for 'y' is already present. Adding another scale for 'y', which
## will replace the existing scale.

Difference of means
## Joining, by = "sex"


The ratio of the intervals is \(R \approx 0.25\) (and \(n = 39)\).
Regression slope


The ratio of the vertical intervals is about \(R \approx 0.65\) (and \(n = 39\)).
Difference in proportions
Fill in an example, say, domhand versus sex
Slope of proportion
Fill in an example, say sex versus width.
Multiple regression